Differential Equations in Abstract Spaces by V. Lakshmikantham

By V. Lakshmikantham

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Let YE C [ S , Y ] and F-differentiable at each point of a convex set V c S. Then for any pair of points x, x + h E V Ilf(x+h) -f(x)ll < llhll SUP Ilf‘(x+WII. 7. Notes Most of the results of this chapter are taken from Vainberg [75] and Hille and Phillips [28]. More details about Bochner integrals and abstract functions can be found in Hille and Phillips [28]. 4 is due to Dieudonne [16]. For other mean value theorems used in connection with differential equations the reader is referred to Aziz and Diaz [S].

We first prove that the function I( T(t)ll is bounded for t in a finite interval [0, t o ] . If not, there exists a sequence t , t* E [0, to] for t,, E [0, to] such that IIT(t,,)ll co while 11 T(r*)ll is a finite number. For every s E X we have T(t,,)s-, T ( t * ) xas n-+ 00. Hence sup. IIT(t,,)xll < 00 for each x E X. By the uniform boundedness principle (see Appendix VI) we conclude that supn IIT(t,,)II co, which is a contradiction. 1, we have that w0 = (log I(T(t)ll)/texists and is a finite number or - co.

These two facts imply that D ( A ) is dense in X . Finally we show that A on D ( A ) is a closed operator. Let x, E D ( A ) with n = 1,2,. , x, = x and limn+mA x , = y . We must prove that x E D ( A ) and that y = A x . By (c) and the fact that T ( s ) A x , + T ( s ) y uniformly we get 1; T ( s ) s d sE D ( A ) for T ( r ) x - x = lim [ T ( t ) x , - x , ] n- m = lim L T ( s ) A x , ds n- = m k ( s ) y ds. Because of this and (d) we have lim A , x = lim f40+ r+o+ t-' l T ( s ) y ds = T(O)Y = Y , which proves that x E D ( A ) and A x = y .

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